New Type of Conserved Quantities of Lie Symmetry for Nonholonomic Mechanical Systems in Phase Space

The new types of conserved quantities,which are directly induced by Lie symmetry of nonholonomicmechanical systems in phase space,are studied.Firstly,the criterion of the weak Lie symmetry and the strong Liesymmetry are given.Secondly,the conditions of existence of the new type of conserved quantities induced by the weakLie symmetry and the strong Lie symmetry directly are obtained,and their form is presented.Finally,an Appell-Hamelexample is discussed to further illustrate the applications of the results.

Lie Symmetry and Conserved Quantities for Mechanical-Electrical Systems

在这篇论文，我们为一个机械电的系统学习谎言对称和保存数量。为这个系统的谎言对称的标准被给。概括 Hojman 保存了数量并且概括了保存数量从谎言对称推出了的 Lutzky 因为系统被获得。一个例子被举说明结果。

Noether-Lie symmetry and conserved quantities of mechanical system in phase space

In this paper, a new kind of symmetry and its conserved quantities of a mechanical system in phase space are studied. The definition of this new symmetry, i.e. a Noether-Lie symmetry, is presented, and the criterion of this symmetry is also given. The Noether conserved quantity and the generalized Hojman conserved quantity of the Noether Lie symmetry of the system are obtained. The Noether-Lie symmetry contains the Noether symmetry and the Lie symmetry, and has more generalized significance.

On Form Invariance,Lie Symmetry and Three Kinds of Conserved Quantities of Generalized Lagrange's Equations

In this paper,the form invariance and the Lie symmetry of Lagrange's equations for nonconservativesystem in generalized classical mechanics under the infinitesimal transformations of group are studied,and the Noether'sconserved quantity,the new form conserved quantity,and the Hojman's conserved quantity of system are derived fromthem.Finally,an example is given to illustrate the application of the result.

The Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems

This paper investigates the Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems. The variational principle of discrete mechanics, from which discrete motion equations of systems are deduced, is generalized to the case of including the time variational. The requirement for an invariant group transformation is defined to be the Lie symmetry and the criterion when the Noether conserved quantities may be obtained from Lie symmetries is also presented. An example is discussed for applications of the results.

The Lie symmetries and Noether conserved quantities of discrete mechanical systems with variable mass

This paper studies the Lie symmetries and Noether conserved quantities of discrete mechanical systems with variable mass. The discrete Euler-Lagrange equation and energy evolution equation are derived by using a total variational principle. The invariance of discrete equations under infinitesimal transformation groups is defined to be Lie symmetry. The condition of obtaining the Noether conserved quantities from the Lie symmetries is also presented. An example is discussed for applications of the results.

LIE SYMMETRIES AND CONSERVED QUANTITIES OF SECOND-ORDER NONHOLONOMIC MECHANICAL SYSTEM

The Lie symmetries and the conserved quantities of the second-order nonholonomic mechanical system are studied. Firstly, by using the invariance of the differential equation of motion under the infinitesimal tran, formations, the determining equations and the restriction equations of the Lie symmetries of the system are established, and the structure equation and the conservative quantities of the Lie symmetries are obtained. Secondly, the inverse problems of the Lie symmetries are studied. Finally, an example is given to illustrate the application of the result.

LIE SYMMETRIES AND CONSERVED QUANTITIES OF ROTATIONAL RELATIVISTIC SYSTEMS

The Lie symmetries and conserved quantities of the rotational relativistic holonomic and nonholonomic systems were studied. By defining the infinitesimal transformations' generators and by using the invariance of the differential equations under the infinitesimal transformations, the determining equations of Lie symmetries for the rotational relativistic mechanical systems are established. The structure equations and the forms of conserved quantities are obtained. An example to illustrate the application of the results is given.

Lie symmetry and Hojman conserved quantity of Nambu system

This paper studies the Lie symmetry and Hojman conserved quantity of the Nambu system. The determining equations of Lie symmetry for the system are given. The conditions for existence and the form of the Hojman conserved quantity led by the Lie symmetry for the system are obtained. Finally, an example is given to illustrate the application of the results.

Lie Symmetry and Non-Noether Conserved Quantity for Hamiltonian Systems

A non-Noether conserved quantity for the Hamiltonian system is studied. A particular infinitesimal transformation is given and the determining equations of Lie symmetry are established. An existence theorem of the non-Noether conserved quantity is obtained. An example is given to illustrate the application of the result.

New conserved quantities of Noether-Mei symmetry of mechanical system in phase space

This paper studies two new types of conserved quantities deduced by Noether Mei symmetry of mechanical system in phase space. The definition and criterion of Noether Mei symmetry for the system are given. A coordination function is introduced, and the conditions under which the Noether- Mei symmetry leads to the two types of conserved quantities and the forms of the two types of conserved quantities are obtained. An illustrative example is given. The coordination function can be selected according to the demand for finding the gauge function, and the choice of the coordination function has multiformity, so more conserved quantities deduced from Noether Mei symmetry of mechanical system can be obtained.

New Conserved Quantities of Noether-Mei Symmetry for Nonholonomic Mechanical System

Two new types of conserved quantities deduced by Noether-Mei symmetry of nonholonomic mechanicalsystem are studied.The definition and criterion of Noether-Mei symmetry for the system are given.A coordinationfunction is introduced,and the conditions under which the Noether-Mei symmetry leads to the two types of conservedquantities and the forms of the two types of conserved quantities are obtained.An illustrative example is given.Thecoordination function can be selected according to the demand for finding the gauge function,and the choice of thecoordination function has multiformity,so more conserved quantities deduced from Noether-Mei symmetry of mechanicalsystem can be obtained.

Two New Types of Conserved Quantities Deduced from Noether Symmetry for Nonholonomic Mechanical System

为 nonholonomic 机械系统，数量从系统的 Noether 对称推出了的保存数量和新概括 Hojman 保存了的概括 Mei 被学习。为系统的 Noether 对称的标准方程被得到。Noether 对称能在下面导致二新保存数量的条件被介绍，保存数量的形式被获得。最后，一个例子被给说明结果的申请。

Noether-Lie Symmetry of Generalized Classical Mechanical Systems

在这篇论文， Noether 谎言对称和概括古典机械系统的保存数量被学习。定义和为在组的一般无穷小的转变下面的系统的 Noether 谎言对称的标准被给。Noether 保存了数量， Hojman 保存了从 Noether 谎言对称推出的数量被获得。一个例子被给说明结果的申请。

Lie-Mei symmetry and conserved quantities of the Rosenberg problem

The Rosenberg problem is a typical but not too complicated problem of nonholonomic mechanical systems. The Lie-Mei symmetry and the conserved quantities of the Rosenberg problem are studied. For the Rosenberg problem, the Lie and the Mei symmetries for the equation are obtained, the conserved quantities are deduced from them and then the definition and the criterion for the Lie-Mei symmetry of the Rosenberg problem are derived. Finally, the Hojman conserved quantity and the Mei conserved quantity are deduced from the Lie--Mei symmetry.

Two New Types of Conserved Quantities of Mei Symmetry for Holonomic Mechanical System

保存数量的二种新类型直接由 holonomic 的 Mei 对称推断机械系统被学习。为 holonomic 系统的 Mei 对称的定义和标准被给。协作功能被介绍， Mei 对称能直接在下面导致保存数量的二种类型和保存数量的二种类型的形式的条件被获得。一个解说性的例子被给。结果显示协作功能能根据计量器功能的需求适当地被选择，从而，计量器功能能更容易被发现。而且，后来，协作功能的选择有多形，为 holonomic 的 Mei 对称的更多保存数量机械系统能被获得。

Lie symmetry and conserved quantity of a system of first-order differential equations

This paper focuses on studying the Lie symmetry and a conserved quantity of a system of first-order differential equations. The determining equations of the Lie symmetry for a system of first-order differential equations, from which a kind of conserved quantity is deduced, are presented. And their general conclusion is applied to a Hamilton system, a Birkhoff system and a generalized Hamilton system. Two examples are given to illustrate the application of the results.